A Positive Solution for a Weighted p-Laplace Equation with Hardy–Sobolev’s Critical Exponent

Abstract

Here, considering \(-\infty<a<\frac{N-p}{p}\), \(a\le e\le a+1\), \(d=1+a-e\) and \(p^*:=p^*(a,e)=\frac{Np}{N-dp}\), the existence of positive solution of a weighted p-Laplace equation involving vanishing potentials

$$\begin{aligned} -\Delta _{ap}u+V(x)|x|^{-ep^*}|u|^{p-2}u=|x|^{-ep^*}f(u) \end{aligned}$$

in \({\mathbb {R}}^N\) is proved, where the potential V can vanish at infinity with exponential decay and f is a function with subcritical growth of class \(C^1\). We use Del Pino & Felmer’s arguments to overcome the lack of compactness and the Moser iteration method with Caffarelli–Kohn–Nirenberg inequality to obtain estimates of the solution in \( L^{\infty }({\mathbb {R}}^N). \)